Question

Exercise 4: Fine structure of hydrogenic atoms a) Consider a Hamiltonian H-Ho + λΗ. with Mr a small perturbation. Show that in (non-degenerate) perturbation theory the first order correction to the unperturbed, discrete energy level E(Holis given by and the second order by b) Apply this to evaluate the first order corrections to the energy levels (the so-called fine structure) of a hydrogenic atom, that arise due to relativistic corrections. Confirm that the answer for the total first order correction is ri 7L where α-e2/(4πε0he) The Hamiltonian is written in this case as (taking μ-m) p2 Ze2 TEO)T 8m3c 1 1dVr).s L-S 22m2c2r dr One can use the following results 4A7 a떼 .hz? me- nim nlm m-ao Z2 Z3 (0) 1 Z3 Estimate the order of magnitude of the corrections.

Answer 1

o) Given that. Ho+ ConSielex a Hamiltonian AH a Smal pestusbaliop. In Pertusbtion -theoty the first oxder Corsi on to th unpaubed, cks coete encxy levrd Consides a hamdtoniog to be tine intpend o tn pend as

(2i mt-n

caci g.m am rn InLm 9mと amc no